Developments on Shape Optimization at CIMNE

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Developments on Shape Optimization at CIMNE
October 2006
www.cimne.com
Advanced modelling techniques for aerospace SMEs
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Shape parametrization
Design variables Coordinates of some definition
points
GeometryB-spline. Definition points r(i)
B-spline expressionin terms of the coordinates
of polygon definition points ri.
Polygon definition points vector, RObtained
solving VNR(V ? imposed conditions at r(i))
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Shape parametrization
Design variables shape parameters (example of
FANTASTIC ship hull)
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Shape parametrization
Design variables deformation of patches defined
with a C1 continuity interpolation function over
the bulb of a ship hull
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Mesh generation and quality aspects
Shape optimization problem
f objective function x vector of design
variables g set of restrictions

• Deterministic methods
• Evolutionary algorithms

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Mesh generation and quality aspects
Evolutionary methods involves the analysis (FEM)
of many different designs.
Mesh Generation
Influence of mesh generation

• Total computational cost of optimizationclosely
  related to FE analysis cost per design.
• Bad quality of FE analysis
• Introduce noise in the convergence
• Possible bad final solution.

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Mesh generation and quality aspects
Classical strategies for meshing each individual

• Adapt a single existing mesh to all the different
  geometries.
• Existing strategies allow adapting an existing
  mesh for very big geometry modifications
  preventing too much distortion.
• Cheapest strategy
• No control of the discretization error.
• Classical adaptive remeshing for the analysis of
  each design.
• Good quality of results of each design
• High computational cost (each design is computed
  more than once)

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Adaption of a mesh to the boundary shape
modifications
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Generation of an adapted mesh for each design in
one step using error sensitivity analysys

• Mesh adaptivity based on Shape sensitivity
  analysis

Low cost control of discretization error
h-adapted mesh for 1st individual
h-adaptive analysis of representative
h-adapted mesh for 2nd individual
Final h-adapted mesh of representative
h-adapted mesh for 3rd individual
Classical sensitivity analysis
h-adapted mesh for Pth individual
Projection parameters (sensitivity of nodal
coordinates and error indicator)
Projection to individuals
in one-step !!
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Generation of an adapted mesh for each design in
one step using error sensitivity analysys

• Mesh Generator
• Advancing front method
• Background mesh defining the size d at each point.

• Mesh Sensitivity
• Boundary nodal points obtained by the B-spline
  sensitivity analysis.
• Internal nodal points spring analogy (fixed
  number of smoothing cycles)

Mesh sensitivity
Smoothing of nodal coordinates
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Parameterization of the problem
Design variables Coordinates of some definition
points
GeometryB-spline. Definition points r(i)
B-spline expressionin terms of the coordinates
of polygon definition points ri.
Polygon definition points vector, RObtained
solving VNR(V ? imposed conditions at r(i))
Sensitivity analysis of the system of equations
Sensitivity analysis of the B-spline expression
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Mesh generation and mesh sensitivity

• Mesh Generator
• Advancing front method
• Background mesh defining the size d at each point.

• Mesh Sensitivity
• Boundary nodal points obtained by the B-spline
  sensitivity analysis.
• Internal nodal points spring analogy (fixed
  number of smoothing cycles)

Mesh sensitivity
Smoothing of nodal coordinates
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Finite element analysis
Solution of standard elliptic equations
Discretization
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Error estimation
Estimation in energy norm of the error
ZZ-estimator
Stress recovery Global least squares smoothing
Approximation of total energy norm
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Sensitivity analysis of the error estimator
Discrete-Analytical method Discretized
model (element integral expressions) are
analytically differentiated
with
Sensitivities of - displacements- strains -
stresses
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Sensitivity analysis of the error estimator
Sensitivities of smoothed stresses
Sensitivities of error estimator
Sensitivities of the strain energy
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The used evolutionary algorithm
Evolutionary algorithm classical Differential
Evolution (Storn Price).
Parameter vector of i-th individual of
generation t
For each individual, a new trial vector is
created by setting some of the parameters upj(t)
to

• Parameters to be modified and individuals q, r, s
  are randomly selected
• The new vector up(t) replaces xp(t) if it yields
  a higher fitness.
• Non accomplished restrictions integrated in
  objective function using a penalty approach.

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Projection to each design and definition of the
adapted mesh
Representative of population
pth individual of population
Projection using shape sensitivityanalysis
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Pipe under internal pressure
4 design variables Circular internal shape P0.9
MPa svm ? 2 MPa ees lt 1.0 30
individuals/generation

• Optimal analytical solution for external surface
• Circular shape Ropt 10.66666
• Cross section area Aopt 69.725903

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Pipe under internal pressure
185 generations 30 individuals/generation only 3
individuals required additional remeshing
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Pipe under internal pressure
0.46
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Gravity Dam
Optimization of internal boundary 10 desing
variables svm ? 2.75 MPa ees lt 3.0 30
individuals/generation
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Gravity Dam
Original Individual
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Gravity Dam
Original Individual
120 generations 30 individuals/generation only 5
individuals required additional remeshing
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Gravity Dam
Reference mesh
Average Individual in Generation 28
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Fly-wheel
60 design variables 8 independent design
variables svm ? 100 MPa ees lt 5.0 15
individuals/generation
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Fly-wheel
300 generations 15 individuals/generation Weight
reduction 1.53 ? 1.445 kg (0.25 ? 0.17 in the
design area) (Deterministic 1.53 ? 1.45 kg)
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Conclusions

• A strategy for integrating h-adaptive remeshing
  into evolutionary optimization processes has been
  developed and tested
• Adapted meshes for each design are obtained by
  projection from a reference individual using
  shape sensitivity analysis
• Quality control of the analysis of each design is
  ensured
• Full adaptive remeshing over each design is
  avoided
• Low computational cost (only one analysis per
  design)
• Numerical tests show
• The strategy does not affect the convergence of
  the optimization process
• Good evaluation of the objective function and the
  constraints for each different design is ensured

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Thank you very much
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